*Coordinate Transformations with t*

*r* = space coordinate, *t* = time coordinate, *v* = velocity, *u* = pace

Galileian transformation

speed: *r´* = *r* – *vt* and *t*′ = *t*, pace: *r´* = *r* – *t/u* and *t*′ = *t*.

Co-Galileian transformation

speed: *t´* = *t* – *r/v* and *r*′ = *r*, pace: *t´* = *t* – *ur* and *r*′ = *r*.

Light: *c* := speed of light, *ç* := pace of light.

speed: *r* = *ct* or* r/c = t* and *r′* = *ct′*, or *r′/c* = *t′*,

pace: *çr* = *t* or *r* = *t/ç* and *çr′ *= *t′* or *r′ *= *t′/ç*.

Lorentz transformation

speed: *γ* = (1 – *v*^{2}/*c*^{2})^{–1/2} with *r′* = *γ* (*r − vt*) and *t′* = *γ* (*t* – *rv/c²*),

pace: *γ* = (1 – *ç*^{2}/*u*^{2})^{–1/2} with *r′* = *γ* (*r* – *t/u*) and *t′* = *γ* (*t* – *rç²/u*),

which applies only if |*v*| < |*c*| or |*u*| > |*ç*|.

Co-Lorentz transformation

speed: *λ* = (1 − *c*^{2}/*v*^{2})^{–1/2} with *t′* = *λ* (*t − r/v*) and *r′* = *λ* (*r* − *t* (*c*^{2}*/v*)),

pace: *λ* = (1 – *u*^{2}/*ç*^{2})^{–1/2} with *t′* = *λ* (*t* – *ur*) and *r′ *= *λ* (*r* – *t* (*u/ç²*)),

which applies only if |*v*| > |*c*| or |*u*| < |*ç*|.

Galilei-Lorentz limit

speed: *c*_{1} → ∞: *γ*_{1} → 1, *t*′ → *t*, *c*_{2} → *c*/2: *γ*_{2} → (1 – 4*v*²/*c*²)^{−1/2} and *t*′ → *γ*_{2} (*t* – 4*r* (*v*/*c*²)).

pace: *ç*_{1} → 0: *γ*_{1} → 1, *t*′ → *t*, *ç*_{2} → 2/*ç*: *γ*_{2} → (1 – (1/4) *ç*²/*u*²)^{−1/2} and *t*′ → *γ*_{2} (*t* – (*r*/4) (*ç*²/*u*)).

Co-Galilei-Lorentz limit

speed: *c*_{1} → 0: *λ*_{1} → 1, *r*′ → *r*, *c*_{2} → 2/c: *λ*_{2} → (1 – (1/4) *c*²/*v*²)^{−1/2} and *r*′ → *λ*_{2} (*r* – (*t*/4) (*c*²/*v*)).

pace: *ç*_{1} → ∞: *λ*_{1} → 1, *r*′ → *r*, *ç*_{2} → *ç*/2: *λ*_{2} → (1 – 4*u*²/*ç*²)^{−1/2} and *r*′ → *λ*_{2} (*r* – 4*t* (*u*/*ç*²)).

If |*v*| = |*c*|, then *r′ = r* and *t′ = t*.

*Coordinate Transformations with τ*

*r* = space coordinate, *t* = time coordinate, *v* = velocity, *u* = pace

*c* := speed of light, *ç* := pace of light.

*τ* := *ct τ* := *t*/*ç*

Galileian transformation

speed: *r´* = *r* – *τ* (*v*/*c*) and *τ*′ = *τ*, pace: *r´* = *r* – *τ (ç*/*u*) and *τ*′ = *τ*.

Co-Galileian transformation

speed: *τ´* = *τ* – *r* (*c*/*v*) and *r*′ = *r*, pace: *τ´* = *τ* – *r* (*u*/*ç*) and *r*′ = *r*.

Light: *r* = *τ* and *r′* = *τ′*

Lorentz transformation

speed: *γ* = (1 – *v*^{2}/*c*^{2})^{–1/2} with *r′* = *γ* (*r − **τ* (*v*/*c*)) and *τ′* = *γ* (*τ* – *r* (*v*/*c*)),

pace: *γ* = (1 – *ç*^{2}/*u*^{2})^{–1/2} with *r′* = *γ* (*r* – *τ* (*ç*/*u*)) and *τ′* = *γ* (*τ* – *r* (*ç*/*u*)),

which applies only if |*v*| < |*c*| or |*u*| > |*ç*|.

Co-Lorentz transformation

speed: *λ* = (1 − *c*^{2}/*v*^{2})^{–1/2} with *τ′* = *λ* (*τ – r* (*c*/*v*)) and *r′* = *λ* (*r* − *τ* (*c*/*v*)),

pace: *λ* = (1 – *u*^{2}/*ç*^{2})^{–1/2} with *τ′* = *λ* (*τ* – *r* (*u*/*ç*)) and *r′ *= *λ* (*r* – *τ* (*u*/*ç*)),

which applies only if |*v*| > |*c*| or |*u*| < |*ç*|.

Galilei-Lorentz limit

speed: *c*_{1} → ∞: *γ*_{1} → 1, *τ*′ → *τ*, *c*_{2} → *c*/2: *γ*_{2} → (1 – 4*v*²/*c*²)^{−1/2} and *τ*′ → *γ*_{2} (*τ* – 4*r* (*v*/*c*)).

pace: *ç*_{1} → 0: *γ*_{1} → 1, *τ*′ → *τ*, *ç*_{2} → 2/*ç*: *γ*_{2} → (1 – (1/4) *ç*²/*u*²)^{−1/2} and *τ*′ → *γ*_{2} (*τ* – (*r*/4) (*ç*/*u*)).

Co-Galilei-Lorentz limit

speed: *c*_{1} → 0: *λ*_{1} → 1, *r*′ → *r*, *c*_{2} → 2/c: *λ*_{2} → (1 – (1/4) *c*²/*v*²)^{−1/2} and *r*′ → *λ*_{2} (*r* – (*τ*/4) (*c*/*v*)).

pace: *ç*_{1} → ∞: *λ*_{1} → 1, *r*′ → *r*, *ç*_{2} → *ç*/2: *λ*_{2} → (1 – 4*u*²/*ç*²)^{−1/2} and *r*′ → *λ*_{2} (*r* – 4*τ* (*u*/*ç*)).

If |*v*| = |*c*|, then *r′ = r* and *τ′ = τ*.

*Coordinate Transformations with c := 1*

*r* = space coordinate, *t* = time coordinate, *v* = velocity, *u* = pace

Galileian transformation

speed: *r´* = *r* – *vt* and *t*′ = *t*, pace: *r´* = *r* – *t/u* and *t*′ = *t*.

Co-Galileian transformation

speed: *t´* = *t* – *r/v* and *r*′ = *r*, pace: *t´* = *t* – *ur* and *r*′ = *r*.

Light: 1 = *c* := speed of light, 1 = *ç* := pace of light.

speed: *r* = *t* and *r′* = *t′*

pace: *r* = *t* and *r′ *= *t′*.

Lorentz transformation

speed: *γ* = (1 – *v*^{2})^{–1/2} with *r′* = *γ* (*r − vt*) and *t′* = *γ* (*t* – *rv*),

pace: *γ* = (1 – 1/*u*^{2})^{–1/2} with *r′* = *γ* (*r* – *t/u*) and *t′* = *γ* (*t* – *r/u*),

which applies only if |*v*| < 1 or |*u*| > 1.

Co-Lorentz transformation

speed: *λ* = (1 − 1/*v*^{2})^{–1/2} with *t′* = *λ* (*t − r/v*) and *r′* = *λ* (*r* − *t* (1*/v*)),

pace: *λ* = (1 – *u*^{2})^{–1/2} with *t′* = *λ* (*t* – *ur*) and *r′ *= *λ* (*r* – *tu*),

which applies only if |*v*| > 1 or |*u*| < 1.

If |*v*| = 1, then *r′ = r* and *t′ = t*.